The value of ${ }^{47} \mathrm{C}_4+\sum\limits_{\mathrm{j}=1}^5{ }^{(52-\mathrm{j})} \mathrm{C}_3$ is

Derivative of

$y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\ldots \ldots \ldots \ldots \ldots \ldots \infty}}}$ is

The feasible region for the constraints $x-y \geq 0, x-5 y \leq-5, x \geq 0, y \geq 0$ is shown by the figure:

The area of the triangle whose vertices are $i, \omega$ and $\omega^2$ is (Where $\omega$ is a complex cube root of unity other than $1, i$ is an imaginary number)__________ sq.units